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A closed formula for the evaluation of foams 评价泡沫的封闭公式
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-08-22 DOI: 10.4171/qt/139
Louis-Hadrien Robert, E. Wagner
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引用次数: 29
Non-semisimple 3-manifold invariants derived from the Kauffman bracket 由Kauffman括号导出的非半简单三流形不变量
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-07-21 DOI: 10.4171/QT/164
M. Renzi, J. Murakami
{"title":"Non-semisimple 3-manifold invariants derived from the Kauffman bracket","authors":"M. Renzi, J. Murakami","doi":"10.4171/QT/164","DOIUrl":"https://doi.org/10.4171/QT/164","url":null,"abstract":"We recover the family of non-semisimple quantum invariants of closed oriented 3-manifolds associated with the small quantum group of $mathfrak{sl}_2$ using purely combinatorial methods based on Temperley-Lieb algebras and Kauffman bracket polynomials. These invariants can be understood as a first-order extension of Witten-Reshetikhin-Turaev invariants, which can be reformulated following our approach in the case of rational homology spheres.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"30 17 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84731137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Tensor product categorifications, Verma modules and the blob 2-category 张量积分类，Verma模和blob 2类
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-05-13 DOI: 10.4171/QT/156
Abel Lacabanne, Gr'egoire Naisse, P. Vaz
{"title":"Tensor product categorifications, Verma modules and the blob 2-category","authors":"Abel Lacabanne, Gr'egoire Naisse, P. Vaz","doi":"10.4171/QT/156","DOIUrl":"https://doi.org/10.4171/QT/156","url":null,"abstract":"We construct a dg-enhancement of Webster's tensor product algebras that categorifies the tensor product of a universal sl2 Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are two-dimensional. This action intertwines with the categorical action of sl2. From the above we derive a categorification of the blob algebra.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"53 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87478265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 6
Holonomy invariants of links and nonabelian Reidemeister torsion 连杆的完整不变量与非abel Reidemeister扭转
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-05-03 DOI: 10.4171/QT/160
Calvin McPhail-Snyder
{"title":"Holonomy invariants of links and nonabelian Reidemeister torsion","authors":"Calvin McPhail-Snyder","doi":"10.4171/QT/160","DOIUrl":"https://doi.org/10.4171/QT/160","url":null,"abstract":"We show that the reduced $mathrm{SL}_2(mathbb{C})$-twisted Burau representation can be obtained from the quantum group $mathcal{U}_q(mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of $mathcal{U}_q(mathfrak{sl}_2)$ satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the $operatorname{SL}_2(mathbb{C})$-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin, and we interpret their invariant as a twisted Conway potential.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"10 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88601129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
A unification of the ADO and colored Jones polynomials of a knot 一个结的ADO和彩色琼斯多项式的统一
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-03-22 DOI: 10.4171/qt/161
Sonny Willetts
{"title":"A unification of the ADO and colored Jones polynomials of a knot","authors":"Sonny Willetts","doi":"10.4171/qt/161","DOIUrl":"https://doi.org/10.4171/qt/161","url":null,"abstract":"In this paper we prove that the family of colored Jones polynomials of a knot in $S^3$ determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored Jones polynomials. On one hand, the first variable $q$ can be evaluated at $2r$ roots of unity with $r in Bbb N^*$ and we obtain the ADO polynomial over the Alexander polynomial. On the other hand, the second variable $A$ evaluated at $A=q^n$ gives the colored Jones polynomials. From this, we exhibit a map sending, for any knot, the family of colored Jones polynomials to the family of ADO polynomials. As a direct application of this fact, we will prove that every ADO polynomial is q-holonomic and is annihilated by the same polynomials as of the colored Jones function. The construction of the unified invariant will use completions of rings and algebra. We will also show how to recover our invariant from Habiro's quantum $mathfrak{sl}_2$ completion studied in arXiv:math/0605313.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"175 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79746984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 16
Homological mirror symmetry for invertible polynomials in two variables 二元可逆多项式的同调镜像对称
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-03-02 DOI: 10.4171/QT/163
Matthew Habermann
{"title":"Homological mirror symmetry for invertible polynomials in two variables","authors":"Matthew Habermann","doi":"10.4171/QT/163","DOIUrl":"https://doi.org/10.4171/QT/163","url":null,"abstract":"In this paper we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the $B$-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and as an application, we prove derived equivalences between certain stacky nodal curves, some of whose connected components have non-trivial generic stabiliser.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"195 1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78328374","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
The FKB invariant is the 3d index FKB不变量是3d索引
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-02-29 DOI: 10.4171/qt/171
S. Garoufalidis, R. Veen
{"title":"The FKB invariant is the 3d index","authors":"S. Garoufalidis, R. Veen","doi":"10.4171/qt/171","DOIUrl":"https://doi.org/10.4171/qt/171","url":null,"abstract":"We identify the q-series associated to an 1-efficient ideal triangulation of a cusped hyperbolic 3-manifold by Frohman and Kania-Bartoszynska with the 3D-index of Dimofte-Gaiotto-Gukov. This implies the topological invariance of the $q$-series of Frohman and Kania-Bartoszynska for cusped hyperbolic 3-manifolds. Conversely, we identify the tetrahedron index of Dimofte-Gaiotto-Gukov as a limit of quantum 6j-symbols.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"32 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72899437","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Factorization homology and 4D TQFT 分解同调与4D TQFT
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-02-20 DOI: 10.4171/QT/159
A. Kirillov, Ying Hong Tham
{"title":"Factorization homology and 4D TQFT","authors":"A. Kirillov, Ying Hong Tham","doi":"10.4171/QT/159","DOIUrl":"https://doi.org/10.4171/QT/159","url":null,"abstract":"In [BK], it is shown that the Turaev-Viro invariants defined for a spherical fusion category $mathcal{A}$ extends to invariants of 3-manifolds with corners. In [Kir], an equivalent formulation for the 2-1 part of the theory (2-manifolds with boundary) is described using the space of \"stringnets with boundary conditions\" as the vector spaces associated to 2-manifolds with boundary. Here we construct a similar theory for the 3-2 part of the 4-3-2 theory in [CY1993].","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"60 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84451492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
Mapping class group actions from Hopf monoids and ribbon graphs 从Hopf monoids和带状图映射类群动作
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-02-10 DOI: 10.4171/QT/158
C. Meusburger, T. Voss
{"title":"Mapping class group actions from Hopf monoids and ribbon graphs","authors":"C. Meusburger, T. Voss","doi":"10.4171/QT/158","DOIUrl":"https://doi.org/10.4171/QT/158","url":null,"abstract":"We show that any pivotal Hopf monoid $H$ in a symmetric monoidal category $mathcal{C}$ gives rise to actions of mapping class groups of oriented surfaces of genus $g geq 1$ with $n geq 1$ boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter-Drinfeld modules over $H$. They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where $mathcal{C}$ is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter-Drinfeld module structure.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"7 Suppl 5 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76610518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Constructing modular categories from orbifold data 从轨道数据构造模块类别
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-02-03 DOI: 10.4171/qt/170
Vincentas Mulevičius, I. Runkel
{"title":"Constructing modular categories from orbifold data","authors":"Vincentas Mulevičius, I. Runkel","doi":"10.4171/qt/170","DOIUrl":"https://doi.org/10.4171/qt/170","url":null,"abstract":"In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $mathbb{A}$ in a modular fusion category $mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $mathbb{A}$ in $mathcal{C}$, we introduce a ribbon category $mathcal{C}_{mathbb{A}}$ and show that it is again a modular fusion category. The definition of $mathcal{C}_{mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $mathbb{A}$ is given by a simple commutative $Delta$-separable Frobenius algebra $A$ in $mathcal{C}$; (ii) when $mathbb{A}$ is an orbifold datum in $mathcal{C} = operatorname{Vect}$, built from a spherical fusion category $mathcal{S}$. We show that in case (i), $mathcal{C}_{mathbb{A}}$ is ribbon-equivalent to the category of local modules of $A$, and in case (ii), to the Drinfeld centre of $mathcal{S}$. The category $mathcal{C}_{mathbb{A}}$ thus unifies these two constructions into a single algebraic setting.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"6 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2020-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88325039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 10
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